Question: Simplify and expand the following expression: $ \dfrac{3}{t - 4}- \dfrac{4}{3t - 12}+ \dfrac{2t}{t^2 - 8t + 16} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $3$ out of denominator in the second term: $ \dfrac{4}{3t - 12} = \dfrac{4}{3(t - 4)}$ We can factor the quadratic in the third term: $ \dfrac{2t}{t^2 - 8t + 16} = \dfrac{2t}{(t - 4)(t - 4)}$ Now we have: $ \dfrac{3}{t - 4}- \dfrac{4}{3(t - 4)}+ \dfrac{2t}{(t - 4)(t - 4)} $ The least common multiple of the denominators is: $ (t - 4)(t - 4)$ In order to get the first term over $(t - 4)(t - 4)$ , multiply by $\dfrac{3(t - 4)}{3(t - 4)}$ $ \dfrac{3}{t - 4} \times \dfrac{3(t - 4)}{3(t - 4)} = \dfrac{9(t - 4)}{(t - 4)(t - 4)} $ In order to get the second term over $(t - 4)(t - 4)$ , multiply by $\dfrac{t - 4}{t - 4}$ $ \dfrac{4}{3(t - 4)} \times \dfrac{t - 4}{t - 4} = \dfrac{4(t - 4)}{(t - 4)(t - 4)} $ In order to get the third term over $(t - 4)(t - 4)$ , multiply by $\dfrac{3}{3}$ $ \dfrac{2t}{(t - 4)(t - 4)} \times \dfrac{3}{3} = \dfrac{6t}{(t - 4)(t - 4)} $ Now we have: $ \dfrac{9(t - 4)}{(t - 4)(t - 4)} - \dfrac{4(t - 4)}{(t - 4)(t - 4)} + \dfrac{6t}{(t - 4)(t - 4)} $ $ = \dfrac{ 9(t - 4) - 4(t - 4) + 6t} {(t - 4)(t - 4)} $ Expand: $ = \dfrac{9t - 36 - 4t + 16 + 6t}{3t^2 - 24t + 48} $ $ = \dfrac{11t - 20}{3t^2 - 24t + 48}$